Page 10 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

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Unit 1: Bessel s Functions

Since the first three terms are independent of z, therefore the fourth term must also be independent Notes
of z. Let it be a constant c, so that

2
1 d Z
Z dz 2 = c
2
d Z
Or 2 = cZ ...(7)
dz
Similarly, the third term in equation (6) must be free from i.e.

1 d 2
= d
d 2
d 2
Or 2 = d ...(8)
d
With the help of (7) and (8) equation (6) becomes

2
1 d R 1 dR 1
d c = 0
R dr 2 r dr r 2
2
2 d R dR 2
or r 2 r (d cr )R = 0 ...(9)
dr dr
Let us put kr = x, so that

dR dR
= k
dr dx
2
2
d R 2 d R
= k
dr 2 dx 2
By putting these values in (9), we get
2
2 2 d R dr cx 2
k r 2 kr d 2 R = 0
dx dx k
2
2
Putting c = k and D = n , we get
2
2 d R dR 2 2
x 2 x (x n ) R = 0
dx dx
Again put R = y we have
2
2 d y dy 2 2
x 2 x (x n ) y = 0
dx dx
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This is Bessel s differential equation. The solution of this equation is called cylindrical function
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or Bessel s function of order n, denoted as J (x).
n
In this unit we shall be using certain properties of gamma function (x):
(i) (n) is defined by the integral

x n
(n) = e x 1 dx , n 0
0

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